Method and Apparatus for Beamforming with Coupled Antennas

1. A method in a transmitting device, the method comprising: transforming a first channel matrix H into a transformed second channel matrix P −T Source H, where the first channel matrix H is a channel from a transmitter array of antennas of the transmitting device to at least one receiver antenna of a receiving device, and where a transformation matrix P −T Source is a function of a transmitter source impedance and an impedance matrix of the transmitter array of antennas; determining a precoding matrix W that maximizes the capacity for the transformed second channel matrix P −T Source H subject to a power constraint of the precoding matrix W; converting the channel capacity maximizing precoding matrix W for the transformed second channel matrix into an optimal precoding matrix V for the first channel matrix; receiving a signal for transmission; applying the optimal precoding matrix V to the signal to generate a precoded signal for transmission over a physical channel; and transmitting the precoded signal.

2. The method according to claim 1 , wherein each column vector of the precoding matrix W is a left singular vector of the transformed second channel matrix P −T Source H.

3. The method according to claim 1 , wherein the transformation matrix P −T Source for the transformed second channel matrix P −T Source H is an inverse transpose of a right factor of a factorization of a Hermitian and non-negative definite matrix.

4. The method according to claim 3 , wherein the transformed second channel matrix is a product of the transformation matrix P −T Source and the first channel matrix H.

5. The method according to claim 3 , wherein the Hermitian and non -negative definite matrix is a function of a source model of the transmitter, the transmitter source impedance of the transmitting device, and the impedance matrix of the transmitter array of antennas.

8. The method according to claim 1 , wherein the signal comprises a vector signal, wherein columns of the optimal precoding matrix V include a plurality of precoding vectors, and wherein applying comprises multiplying the vector signal by the optimal precoding matrix V to generate a precoded signal vector for transmission over a physical channel.

9. The method according to claim 1 , wherein the transmitter array of antennas are mutually coupled in that voltage or current applied to one antenna element induces a voltage or current on another antenna element in the transmitter array of antennas.

10. The method according to claim 1 , further comprising ascertaining channel measurements of a channel between the transmitting device and the receiving device, wherein the first channel matrix H is based on the channel measurements of the channel between the transmitting device and the receiving device.

11. An apparatus comprising: a transmitter array of antennas; a transceiver coupled to the transmitter array of antennas; and a controller coupled to the transceiver, the controller configured to transform a first channel matrix H into a transformed second channel matrix P −T Source H, where the first channel matrix H is a channel from the transmitter array of antennas to at least one receiver antenna of a receiving device, and where a transformation matrix P −T Source is a function of a transmitter source impedance and an impedance matrix of the transmitter array of antennas, determine a precoding matrix W that maximizes the capacity for the transformed second channel matrix P −T Source H subject to a power constraint of the precoding matrix W, convert the channel capacity maximizing precoding matrix W for the transformed second channel matrix into an optimal precoding matrix V for the first channel matrix, receive a signal for transmission, and apply the optimal precoding matrix V to the signal to generate a precoded signal for transmission over a physical channel, wherein the transceiver is configured to transmit the precoded signal over the physical channel via the transmitter array of antennas.

12. The apparatus according to claim 11 , wherein each column vector of the precoding matrix W is a left singular vector of the transformed second channel matrix P −T Source H.

13. The apparatus according to claim 11 , wherein the transformation matrix P −T Source for the transformed second channel matrix P −T Source H is an inverse transpose of a right factor of a factorization of a Hermitian and non -negative definite matrix.

14. The apparatus according to claim 13 , wherein the transformed second channel matrix P −T Source H is a product of the transformation matrix P −T Source and the first channel matrix H.

15. The apparatus according to claim 13 , wherein the Hermitian and non -negative definite matrix is a function of a source model of a transmitter of the transceiver, the transmitter source impedance, and the impedance matrix of the transmitter array of antennas.

18. The apparatus according to claim 11 , wherein the signal comprises a vector signal, wherein columns of the optimal precoding matrix V include a plurality of precoding vectors, and wherein the controller is configured to apply the optimal precoding matrix V by multiplying the vector signal by the optimal precoding matrix V to generate a precoded signal vector for transmission over a physical channel.

19. The apparatus according to claim 11 , wherein the transmitter array of antennas are mutually coupled in that one of a voltage and current applied to one antenna element induces a voltage or current on another antenna element in the transmitter array of antennas.

20. The method according to claim 11 , wherein the controller is configured to ascertain channel measurements of a channel between the transceiver and the receiving device, and wherein the first channel matrix H is based on the channel measurements of the channel between the apparatus and the receiving device.

21. A method in a transmitting device, the method comprising: ascertaining channel measurements of a channel between the transmitting device and a receiving device; transforming a first channel matrix into a transformed second channel matrix, where the first channel matrix is a channel from a transmitter array of antennas of the transmitting device to at least one receiver antenna of the receiving device, where the first channel matrix is based on the channel measurements of the channel between the transmitting device and the receiving device, where the transmitter array of antennas are mutually coupled in that voltage or current applied to one antenna element induces a voltage or current on another antenna element in the transmitter array of antennas, and where a transformation matrix for the transforming is a function of a transmitter source impedance and an impedance matrix of the transmitter array of antennas; determining a first precoding matrix that maximizes the capacity for the transformed second channel matrix subject to a power constraint of the first precoding matrix; converting the channel capacity maximizing first precoding matrix for the transformed second channel matrix into a second precoding matrix for the first channel matrix; receiving a signal for transmission; applying the second precoding matrix to the signal to generate a precoded signal for transmission over a physical channel; and transmitting the precoded signal.